As soon as the travelling crate was opened and the shroud of white tissue paper carefully peeled away, it was clear there was damage to the dark blue coat: a hole in the left shoulder, and some of the gold braid on the epaulette torn away. The damage happened more than two centuries ago, and the coat's arrival in France was one of the most unusual days in the history of the National Maritime Museum, in Greenwich, south London, and the Musée de l'Armée, at Les Invalides in Paris.

"I think it's a wonder," said Emelie Robbe, a curator of the Paris museum's new exhibition on Napoleon and Europe. "It is astonishing that it should be here."

The coat, an undress uniform of the Royal Navy, already slightly old fashioned when it was made in the late 18th century, had never left England since 1805, when it came back in a sea chest on the same ship that carried the body of Horatio Nelson preserved in a barrel of brandy. It has now voyaged again, through the Channel tunnel, into the heart of his enemy's empire.

The coat has been viewed as a sacred relic of British history, soaked in blood on the day that saw one of the greatest naval victories, at Trafalgar, over the combined fleets of France and Spain, and the disaster of the death of the vice-admiral. The hole in the shoulder was pierced by the musket ball of a French sniper perched high in the rigging of the Redoubtable.

The sniper's bullet passed through Nelson's chest and lodged near his spine, paralysing him and leaving him drowning slowly in his own blood. Amy Miller, a costume curator at Greenwich who escorted the uniform to Paris to see it safely installed, said the wound would probably be beyond surgery even now but, in 1805, Nelson knew instantly that he was finished. He lived just long enough to learn that the battle was won.

The exhibition describes Trafalgar as une victoire écrasante – a crushing victory – for the British fleet.

The blood on the sleeve and the hem of the coat is another grim reminder of the savagery of naval warfare. Nelson's secretary, John Scott, was shot at his side an hour before Nelson himself, whose coat got stained as he stooped to find Scott dying on the deck at his feet.

The uniform was sent back to England, with his blood-soaked silk stockings and the breeches that had been cut off to spare him pain when he was carried below deck for hopeless treatment, and given to his brother William.

His mistress, Emma Hamilton – excluded from all the ceremonies, including the funeral – wrote to the family begging for a loan. A neighbour's child, Lionel Goldsmith, remembered all his life being taken to see the uniform laid out on the bed she had shared with Nelson at their home in Merton, south London.

Hamilton soon got into dire financial trouble, and just before emigrating to Boulogne, where she died, she pledged it to one Joshua Smith, to settle a debt. When Smith's widow planned to sell it to a waxworks museum, Prince Albert instead bought it for £150, and presented it in 1845 to the Royal Hospital in Greenwich. Since then, it has never moved further than a few hundred yards, being exhibited for decades in the vast Painted Hall dining room before being transferred to the National Maritime Museum, across the road.

In 2005 the Paris museum sent spectacular loans to the Maritime's Nelson and Napoleon exhibition, including the uniform Napoleon wore at the battle of Marengo; the reciprocal loan is happening while a new Nelson gallery is created at Greenwich.

"It is an extraordinary object," Miller said. "Uniforms have great power. They are as close as you can get to the body of a historical figure."

Tellingly, she and the conservator Nicky Yates unconsciously referred to the coat as "him" as they moved it carefully from the packing crate into the display case.

The most recent conservation work revealed that the coat – possibly made by a tailor in Portsmouth, where Nelson's flagship, Victory, is still berthed – was already several years old at the time of the battle.

Miller said: "It has been suggested that Nelson wore a uniform with shining gilt decorations and brass buttons on the day of the battle out of vanity or bravado. But that is clearly not true. It was an old, comfortable coat, and very suitable for the day. The decorations were already tarnished in 1805, and he was standing beside Hardy, who was taller and also wearing a coat with brass buttons. So he was not, in fact, an obvious target."

Touchingly, they found that the empty sleeve, for the right arm Nelson had lost in battle eight years earlier, was only partly lined, to save money.

There are oOther uniforms in the exhibition, including those of the emperors of Austria and Russia, are also travelling abroad for the first time. However, Emelie Robbe said the arrival of Nelson's was special.

"It is a really beautiful uniform," she said. "French people are always thinking the French uniforms are the best cut – but this one is a rival."

Nelson's body took months to get back to England after the battle. It was carried up the Thames on a barge from the Royal Hospital on 8 January 1806, followed by a flotilla of boats, the riverbanks and bridges packed with spectators. It was buried in St Paul's Cathedral the next day after a great procession, during which the carriage carrying his coffin was dragged through the streets by a party of his sailors.

Napoleon, defeated at Waterloo in 1815 and exiled to St Helena, died in 1821. In 1840, his body was exhumed and was finally buried under the immense golden dome of Les Invalides after a funeral procession watched by an estimated 600,000 people.

Nelson's uniform now stands among those of three emperors, in a city he never visited in life.

" We have placed the uniform of Napoleon so that he looks towards the uniforms of the emperors of Austria and Russia, who he absolutely crushed," Robbe explained. "But at his back, there was always the shadow of Nelson, and so we have placed him there."

• This article was amended on 28 March 2013. The costume curator at Greenwich is Amy Miller, not Wilson as the original said. The original also referred to the English fleet and English history. This has been corrected.